4-th Project Workshop Ageing and Maintenance in Reliability: Modelling and Statistical Inference
The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena
Gianpaolo Pulcini Istituto Motori National Research Council (CNR), Italy
The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
For several highly reliable products, it is often difficult to evaluate their reliability using classical methods based on failure times due to the very low number of failures that are generally observed.
Fortunately, many such products have performance characteristics whose decay or degradation over the operating time can be related to their reliability. Thus, if the degradation phenomenon of these products is observed and adequately modelled, it is possible to accurately evaluate their reliability.
In addition, an accurate modelization of the degradation phenomenon allows one to predict the remaining life, and then we can plan a condition-based maintenance policy, which can be more effective both than an age-based preventive maintenance and than a corrective maintenance policy.
Of course, if the degradation phenomenon is incorrectly modelled, the reliability evaluation, the lifetime prediction and the maintenance optimization can result in inaccurate.
2 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Let {W (t); t 0} denote an increasing, continuous-time, stochastic process. In general, when a unit is subject to a degradation phenomenon, the probability distribution of the
degradation increment W(t,T ) W(t T ) W(t) over the future time interval (t, t T ) depends on the whole history of the degradation process up to the current time t .
When W(t,T ) is assumed to depend only on the present and not on the past history, then the process is said to be Markovian. Within the Markovian processes, we can distinguish the cases where depends (functionally or stochastically):
fw(w) a) only on the interval length T (homogeneous or
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d) on , on the current age t and state wt (age- and t t+T Operating time state-dependent process with dependent increments)
3 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
When the degradation growth of an item depends only on its age and not on the current degradation, a widely accepted model is the non-stationary Gamma process.
The non-stationary Gamma process {W(t), t 0} has the following properties:
(a) the increments W(t,T ) W(t T ) W(t) (t, T 0) over disjoint time intervals (t, t T ) are independent random variables, and
(b) each increment W(t, T ), for all t and T , follows a Gamma distribution with constant scale
parameter 0 and (possibly) time-varying shape parameter (t,T ) (t T ) (t):
(w/ ) (t , T ) 1 f (w|t) exp(w/ ) , w0 (1) W (t , T ) [(t, T )]
() is a non-negative, monotone increasing function for t 0 with (0) 0, a1 (a) 0 z exp(z) dz is the complete Gamma function.
The unusual notation f (w| t) is here used to stress the “functional” dependence on the current age W (t, T )
If (t) t , the process is homogeneous: the increments distribution is independent on the age t .
4 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
A unit subject to the degradation process is generally assumed to fail when its degradation level
exceeds a threshold level wmax . The unit lifetime is then defined as the operating wmax
time needed to exceed wmax .
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The residual reliability is the probability that: x R (x| w ) Pr{W (t,x) w w |t,w } t t max t t t t+x Operating time and, under the Gamma process, is given by:
( w w )/ max t (t ,x) 1 z R (x| w ) exp(z) dz . (2) t t [(t, x)] 0
It depends on the current state wt only through the difference wmax wt .
5 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The mean and variance of the degradation W (t) at time t of the Gamma process are: E{W(t)}(t) and V{W(t)}(t) 2 . (3)
Thus, the variance-to-mean ratio V{W(t)}/ E{W(t)} equals the scale parameter and does not depend on time t .
Note that even in the Inverse Gaussian process (the other commonly used process with positive increments) the ratio V{W (t)}/ E{W (t)} is constant with t
The independent increments property and the assumption of a constant scale parameter make the subsequent mathematics quite tractable, and have encouraged the widespread use of the Gamma process, even without performing the necessary checks on the adequacy of the Gamma process to the degradation phenomenon to analyze.
For example, even if the degradation process is purely age-dependent, the non-stationary Gamma process is not a proper choice when, in absence of heterogeneity and/or measurement errors, there is empirical evidence that the variance-to-mean ratio is not constant.
6 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
A degradation model where the variance-to-mean ratio is not restricted to be constant is the Extended Gamma process, which is defined by: t X (t) (s) dZ (s) , (4) 0 {Z(s), s 0} is a Gamma process with shape parameter (s) and unit scale parameter, (s) is a positive, right continuous, real-valued function, the integration is done with respect to the sample paths of Z (s).
The Extended Gamma process {X (t), t 0} has independent increments, and when (s) is a constant for all s 0, it reduces to the Gamma process.
The mean and variance of W (t) are given by t t E{X (t)} (s) d(s) and V{X (t)} 2 (s) d(s) , (5) 0 0 and hence the variance-to-mean ratio is no longer necessarily constant.
Unfortunately, the distribution function of X (t) is very difficult to treat mathematically.
7 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
A way to overcome the mathematical difficulty of the Extended Gamma process is to discretize the time axis t with (small) time increments of length h .
The time-discretized model W (t) (called the time-discrete Extended Gamma process) is based on the following assumptions:
1. the increments over disjoint time intervals are independent random variables,
2. each “elementary” increment W (t, h)W (th)W (t) over a very small time interval (t, t T ) of length h , follows a Gamma distribution with mean and variance:
E{W(t, h)|t}(t h) (t) V{W(t,h)|t} (t h) (t), (6)
where () and () are differentiable, monotone increasing functions, with (0) 0 and (0) 0.
8 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Then, under the time-discrete Extended Gamma process: a) the Gamma distribution of each increment W(t,h) has time-varying scale and shape parameters: V{W (t, h)} (t h) (t) E2{W (t, h)} [(t h) (t)]2 (t) and (t) (7) E{W (t, h)} (t h) (t) V{W (t, h)} (t h) (t) b) the increment W (t, h) W (t h) W (t) ( 2, 3,...) over the non-elementary time interval (t, t h) of length h is the sum of independent, but not identically distributed Gamma r.v.: W (t, v h) W (t (l 1) h, h) Wl , (8) l1 l1
and each Gamma r.v. Wl has scale and shape parameters which depend on the age t l h at the beginning of the elementary time interval and on the length h : (t l h) [t (l 1) h] {(t l h) [t (l 1) h]}2 (t l h) and (t l h) (9) l (t l h) [t (l 1) h] l (t l h) [t (l 1) h] c) the mean and variance of the degradation level W (t) at time t are: E{W (t)}(t) and V{W(t)} (t) , and hence the variance-to-mean ratio is not necessarily constant with t .
9 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
As h 0, the time-discrete Extended Gamma process tends to the Extended Gamma process with time-varying scale parameter (t) '(t)/'(t) and with d(t) ['(t)]2 / '(t) dt .
Indeed, for such an Extended Gamma process, it results that dX (t) is a Gamma random variable with
'(t) ['(t)]2 E{dX (t)|t} (t) d(t) dt '(t) dt '(t) '(t)
2 2 2 '(t) ['(t)] V{dX (t) |t} (t) d(t) dt '(t) dt . '(t) '(t)
On the other hand, the time-discrete Extended Gamma assumption that the elementary increment W (t, h) is a Gamma random variable yields:
lim E{W (t,h)|t} lim [(t h)(t)] '(t) dt h0 h0 lim V{W (t,h)|t} lim [ (t h) (h)] '(t) dt . h0 h0
Thus, as h 0, the increment W (t,h) converges to dX (t) for each time t , and hence the proposed time-discrete model {W (t), t 0} converges to the Extended Gamma process {X (t), t 0}.
10 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Since the elementary increments Wl are independent Gamma random variables with different scale parameters, the exact density function of the degradation increment W (t, v h) Wl (10) l1 can be obtained, for example, from the following Gamma series:
v l w k1 f (w| t) 1 k exp(w/ ) (11) W (t , h) k 1 l1 l k0 1 ( k)
where: 1 min( l ) , l1l , 0 1, and the k1 1 s coefficients k (k 1, 2,...) satisfy the recurrence relations: k1 l (1 1 / l ) δk1s , k 0, 1,... . k 1 s1 l1
The Gamma series in (11) is very convenient for computational purposes
because the coefficients k (k 0, 1,...) can be easily computed.
For practical purposes, this sum is stopped at the first K 1 terms of the series, where K is determined in order to attain a desired accuracy.
11 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
When all the elementary increments Wl W(t (l 1) h, h) are Gamma distributed with
common scale parameter l (so that the resulting degradation process is just Gamma), then:
1 , 0 1 , and k 0 , for all k 1 .
As a consequence, the pdf of the increment W(t, h) over the non-elementary interval (t, t h)
v l wk1 f (w|t) 1 k exp(w/ ) (12) W (t, h) k ( k) 1 l1 l k0 1 becomes exactly the Gamma density w1 f (w|t) exp(w/ ) (13) W (t, h) ()
with shape parameter l1l (t h)(t) (t, h).
Then, the Gamma process can be viewed as a special case of the time-discrete Extended Gamma process.
12 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Under the time-discrete Extended Gamma process, the conditional residual reliability Rt (x | wt ) is a
non-increasing step function, evaluated at the (discrete) time x h:
( w w )/ v l max t 1 z k1 R (x |w )Pr{W (t, h)w w |t} 1 exp(z) dz . (14) t t max t k ( k) l1 l k0 0
Rt (x |wt ) depends on wt only through the difference wmax wt .
From the reliability expression (14), we can derive both the conditional mean remaining life:
E{X |t,wt } Rt (x |wt ) , (15) 0 and the probability mass function of the remaining lifetime X :
p(x |wt )Pr{X h|t,wt }Rt (x 1 |wt )Rt (x |wt ) , 1, 2,... . (16)
13 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
A number of functional forms can be considered for (t) E{W (t)} and V{W (t)} (t):
the power-law function: a tb (a, b 0)
the exponential function: c [1exp(d t)] (cd 0)
It is immediate to verify that the time-discrete Extended Gamma model includes, as special cases:
(a) the stationary Gamma process, when mean and variance of W (t) are both linear:
E{W (t)}(t) 1 t and V{W (t)} (t) 3 t
(b) the Gamma process with power-law shape parameter (t) tb , when mean and variance are both
power-law functions with common shape parameter 0: (t) 1 t 0 and (t) 3 t 0
(c) the Gamma process with exponential shape parameter (t) [1exp(bt)], when mean and
variance are both exponential functions with common shape parameter 0 :
(t) 1 [1exp(0 t)] and (t) 3 [1exp(0 t)]
In all these cases, the variance-to-mean ratio of W (t) is constant with t and equal to 3 /1.
14 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Numerical application
Real dataset consisting of the sliding wear data (in microns) of four metal alloy specimens, measured at common selected ages up to 500 hundreds of cycles.
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Operating time t [hundreds of cycles] in log scale
15 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
22 1.2 Empirical estimate Empirical estimate
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The time-discrete Extended Gamma process, with h100 cycles and power-law function both for (t) ˆ ˆ ˆ ˆ and (t) (1 6.69, 2 0.161, 3 0.183, and 4 0.687), provides the better fit to the observed data.
The likelihood ratio test rejects the null hypothesis that 2 4 , that is the hypothesis that the process is Gamma with power-law shape parameter (t): 17.50 and p-value = 2.9·10-5.
16 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The property of “independence of increments”, however, confines the use of Gamma processes to
deterioration mechanisms where the degradation growth over the time interval (t, t T ) depends (functionally) on the current age t and not on the (current) state of the unit.
In addition, if there is empirical evidence that, in absence of heterogeneity and/or measurement errors, the variance of the observed process does not increase monotonically with the age t , then the Gamma process (and the Extended Gamma process, as well as any other state-independent degradation process) is not adequate to describe the observed process because the property of “independence of increments” forces the variance of the process to grow monotonically.
Thus, we have to consider a stochastic process able to describe degradation phenomena where the increments (over disjoint time intervals) are not independent (state-dependent processes).
17 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The usual “direct” way of modeling a deterioration phenomenon is in terms of the degradation level as a function of the operating time, say {W(t), t 0}.
However, we can also consider the “inverse” process, that is the time process {T(w), w 0}, i.e., the process of the time T needed for reaching the degradation level w.
When the “direct” process {W(t), t 0} is a positive, increasing pure state-dependent process
(that is, W(t, T ) depends physically only on the current state W(t) wt , and not on the current age t ), then also the “inverse” time process {T(w), w 0} should be a positive and increasing process,
whose increments T(wt ,W ) T(wt W ) T(wt ) over the degradation intervals (wt ,wt W )
depend physically only on the current state wt of the unit, and not on its current age t T(wt ).
Thus, when the “direct” process is a pure state-dependent process, independent on the age, the time process {T(w), w 0} has “independent time increments”, and hence it can be appropriately described by the Gamma process.
18 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Under the assumption that the “inverse” time process {T(w), w 0} is a (possibly) non-stationary Gamma process with shape parameter (w),
then the time increment T(wt ,W ) over the interval (wt ,wt W ) follows a Gamma distribution
with shape parameter (wt ,W ) (wt W ) (wt ) and positive scale parameter :
( / ) ( wt , W ) 1 f ( | w ) T exp( / ) , 0 , (17) T ( wt , W ) T t T T [(wt ,W )]
(w) is a non-negative, monotone increasing function of w, with (0) 0.
The notation f ( |w ) allows us to stress the (functional) dependence on the current degradation level w T (wt , W ) T t t
When (w) is a linear function of w, we have that (wt ,W ) wt and hence the pdf of the time
increment depends only on the length W of the interval, and not on the current state W(t) wt . The time process is then homogeneous.
The mean and variance of the “inverse” process {T(w), w 0} at the degradation level w are:
E{T(w)}(w) and V{T (w)}(w) 2 . (18)
19 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Since the remaining lifetime X is the extra time the unit spends to exceed the threshold level wmax , when starting from the current state wt , the probability distribution of the remaining lifetime is Gamma,
with shape parameter (wt ,wmax wt ) (wmax )(wt ) and scale parameter :
(x/ ) ( wt ,wmax wt ) 1 f X (x | wt ) exp(x/ ) . (20) [(wt ,wmax wt )]
The mean remaining life (MRL) is: E{X |wt } (wt ,wmax wt ) .
The residual reliability of the unit is the probability that the remaining life X exceeds the time x , when
starting from the current degradation level W(t) wt wmax : x (z / ) ( wt ,wmax wt ) 1 R (x| w ) Pr{X x| w }1 exp(z / ) dz , (21) t t t [(w ,w w )] 0 t max t and the reliability of a new unit (t 0 & W(0)0) is:
x (z / ) ( wmax ) 1 R(x) Pr{X x | 0}1 exp(z / ) dz . (22) [(w )] 0 max
20 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Since the process {T(w), w 0} is monotonically increasing, the conditional probability distribution
that the degradation growth W(t,T ) W(t T )W(t) over the time interval (t , t T )
is less than W , given the current state wt , is:
F ( | w ) Pr{W (t, ) | w } Pr{T (w , ) | w } . (23) W (t , T ) W t T W t t W T t
Thus, since is a Gamma process, a degradation process {W(t),t 0}
is said to be an Inverse Gamma process if the conditional Cdf of the degradation growth W(t,T ), given the current state, is given by:
F ( | w ) Pr{T (w , ) | w } W (t , T ) W t t W T t IG[ / ; (w , )] T (t / ) ( wt , w ) 1 1 T t W 1 exp(t / ) dt . (24) [(wt ,W )] 0 [(wt ,W )]
It is evident that the (conditional) distribution of the degradation increment depends on the current
state wt , as well as on the time interval length T , but not on the current time t .
Thus, the Inverse Gamma process can model purely state-dependent degradation phenomena.
21 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
If the shape function (w) is differentiable, the conditional probability density function of W(t,T )
can be obtained by deriving its Cdf with respect to W :
T ( w , ) 1 dFW (t , ) (W | wt ) d (t / ) t W f ( | w ) T exp(t / ) dt . (25) W (t ,T ) W t dW dW 0 [(wt ,W )]
A tractable expression of the conditional pdf of is given by: f ( | w ) W (t ,T ) W t '(w ) (1) k ( / ) k ( wt ,W ) t W IG T ;(w , ) [(w , )] ln( / ) T (26) t W t W T 2 [(wt ,W )] k0 [(wt ,W ) k] k!
where '(a)d(a)/da and (z)dln(z)/dz is the digamma function.
By setting wt 0, we obtain the density function of the degradation level W (t) reached at time t by a new unit: '(w) t (1) k (t / )k ( w) f (w|0) IG ;(w) [(w)] ln(t / ) . (27) W (t ) [(w)] 2 k0 [(w) k] k!
22 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The conditional mean E{W(t,T )|wt} of the degradation growth over the time interval (t ,t T ) of
length T , given the current state wt , follows:
IG[ / ;(w , )] E{W (t, )|w } [1F ( |w )] d T t W d , (28) T t W (t ,T ) W t W [(w , )] W 0 0 t W
and the conditional variance V{W(t,T )|wt} is given by:
V{W (t, )|w } ( E{W (t, )|w })2 f ( |w ) d . (29) T t W T t W (t ,T ) W t W 0
It can be show that the variance V{W (t)} of the degradation process is not constrained to increase monotonically with t , because the IG process has not independent increments.
23 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
For a new unit, the mean degradation level at the time t is:
IG[t / ; (W )] E{W (t)| 0} dW . (30) 0 [(W )] The second derivative w.r.t. is given by:
d 2 E{W (t)|0} (t / ) ( w) 1 (w)1 1 exp(t / ) dw , (31) dt 2 [(w)] t 0
and is negative for small t values, even if (w) w.
Thus, even if the time process {T(w), w 0} is homogeneous (so that E{T(w)} increases linearly), the degradation mean E{W(t)|0} does not increase linearly.
In particular, when , the curve is concave (downwards), especially for small values, and tends to be linear as t increases.
24 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Several functional forms for the (w) function are considered, which determine the behavior of the
mean and variance of the time process {T(w), w 0}:
E{T(w)}(w) and V{T (w)}(w) 2 , (32)
and consequently the behavior of the Inverse Gamma degradation process {W(t),t 0}: a) the exponential (Exp) function: (w) [exp(w/)1], with , , and 0. b) the power-law (PL) function: (w) (w/) , with , 0. c) the logarithmic (Log) function: (w) ln(1w/ ), with 0. d) the linear-exponential (LE) function: (w) {w [1exp(w/)]}, with 0.
25 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Mean degradation curve of the Inverse Gamma processes with power-law (w) (w/) .
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26 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Numerical application
Real dataset consisting in the wear process of 32 cylinder liners of the Diesel engines equipping some
identical cargo ships of the Grimaldi Lines. The data were collected from January 1999 to August 2006.
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Operating time [h]
27 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
5.0 0.6 Empirical estimate Empirical estimate
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MLE IG process t (
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0.0 0.0 0 10000 20000 30000 40000 50000 60000 0 10000 20000 30000 40000 50000 60000 Operating time t [h] Operating time t [h]
Inverse Gamma process on the wear process {W (t), t 0} with (w) [exp( w/ )1] (ˆ 11.19, ˆ 2496.5 hours, ˆ 3.468 mm, ˆ 43.30, AIC80.60) Gamma process on the wear process {W (t), t 0} with (t) [exp(t/ )1] (ˆ 35.72, ˆ 4.076104 hours, ˆ 0.1471 mm, ˆ 22.92, AIC39.84) The above (w) & (t) functions provide the better fit to the data in terms of the Akaike Information Criterion (AIC). The IG process has to be preferred to the Gamma process because the AIC value under the IG process
(AIC80.60) is much smaller than the AIC value under the Gamma process (AIC39.84).
28 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Reliability of a new liner Residual reliability of liner #6
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0.0 s 0.0 E 0 50,000 100,000 150,000 200,000 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 Operating time x [h] Operating time x [h] Inverse Gamma process on the wear process {W (t), t 0} with (w) [exp( w/ )1] Gamma process on the wear process {W (t), t 0} with (t) [exp(t/ )1].
The Gamma process underestimates the reliability and the residual reliability at small operating times
and greatly overestimates the reliability at large x.
29 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
In some cases, however, the observed degradation phenomena can have increments W(t,T ) which depend both on the current age t and on the current state W(t) wt . In these cases, therefore, neither the Gamma process, nor the Inverse Gamma process, can adequately describe the observed process.
A suitable model is the Transformed Gamma (TG) process.
An increasing, continuous-time, degradation process {W (t); t 0} is said to be a TG processes if the (conditional) pdf of W (t,tt) is given by
[g(w ,w)] (t , T ) 1 f (w|t,w ) g'(w w) t exp[g(w ,w)] (33) W (t ,T ) t t t [(t,T )] (t) is a non-negative, monotone increasing function of t (“age function”), with (0) 0, g(w) is a non-negative, monotone increasing and differentiable function of w (“state function”), with g(0) 0,
g'(wt w) is the derivative of g(w) evaluated at wt w,
g(wt ,w) g(wt w) g(wt ) & (t,T ) (t T )(t).
It is evident that, if the age and state functions are not linear, the degradation increment W (t,T )
depends both on the current age t and the current state wt , in addition to the interval length T .
30 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The TG process includes as special case the Gamma process.
1) If the state function is linear, say g(w) w/ , the TG process reduces to the pure age-dependent Gamma process, where the distribution of the degradation increment no more depends on the
current state wt but only (at most) on the current age t .
2) If both g(w) and (t) are linear, the TG process reduces to a Gamma process with stationary increments.
Finally, if the age function is linear, say (t) t /a, the TG process reduces to a pure state-dependent (SD) process (i.e., with increments that, given the current state , are independent of the current age t).
31 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The conditional mean of the degradation increment W (t,T ) over the interval (t, t T ),
given the current age t and degradation level W(t) wt , is: E{W (t, )|t,w } w f (w|t,w ) dw . (34) T t W (t , T ) t 0 The behavior of the state function g(w) has a clear effect on the conditional mean of the degradation increment .
w3 } E{W(t + T)|t,w3}
)
t
(
W
For example, if is convex
l
e E{W(t + T)|t,w2} v w2
e }
l
(downward), the conditional mean
n
o
i
t
a of the increment at the E{W(t + T)|t,w1}
d
a
r } w1 g time t , given that W (t) w , is a e t
D decreasing function of wt . T
t t+T Operating time t
32 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The conditional Cdf of the degradation increment W (t,T ), given the current age t and state wt , is IG[g(w ,w); (t, )] F (w|t,w ) t T (35) W (t , T ) t [(t,T )]
x where IG(x;a) z a 1 exp(z) dz is the incomplete Gamma function. 0
Under the TG process, the conditional residual reliability is given by:
Rt (x|t,wt ) Pr{X x|t,wt } Pr{W (t,x) wmax wt |t,wt }
g ( w ,w w ) IG[g(w ,w w ); (t,x)] t max t z (t ,x) 1 t max t exp(z) dz . (36) [(t,x)] 0 [(t,x)]
and hence it depends both on t and on wt (not only through wmax wt ).
Clearly, the reliability of a new unit is given by: IG[g(w ); (x)] R(x|0,0) max . (37) [(x)]
33 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
If the age function (t) is differentiable, we can obtain the conditional pdf of the remaining lifetime X by deriving the reliability function with respect to x :
g ( w ,w w ) 1 t max t u (t ,x) 1 f (x|t,w ) exp(u) du X t dx [(t,x)] 0
'(t,x) IGg(wt ,wmax wt );(t,x) [(t,x)] ln[g(wt ,wmax wt )] [(t,x)] (38) [g(w ,w w )] (t ,x) t max t F (t,x),(t,x);(t,x)1,(t,x)1;g(w ,w w ) 2 2 2 t max t [(t,x)]
() is the digamma function, '(t,x)d(t,x) dx'(tx),
2 F2 () is the generalized hypergeometric function of order (2, 2).
34 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Since the generalized hypergeometric function 2 F2 () of order (2, 2) can be evaluated by: (t, x)2 [g(w ,w w )]k F t max t (39) 2 2 2 k0 (t, x) k k!
the conditional pdf of the remaining lifetime, given the current age t and state wt , is given by:
'(t x) f X (x|t,wt ) IGg(wt ,wmax wt );(t,x) [(t,x)] k 1 [g(w ,w w )] (t ,x)k [(t,x)]ln[g(w ,w w )] t max t . (40) t max t 2 k0 (t,x)k k!
Note that this sum converged quite quickly in our applications
The mean remaining lifetime can be obtained by numerical integration of the residual reliability IG[g(w ,w w ); (t,x)] E{X |t,w } R (x|t,w ) dx t max t dx . (41) t t t [(t,x)] 0 0
35 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
In order to obtain a fully operative formulation of the TG class of processes, it is necessary to give a functional form to the state function g(w) and to the age function (t).
Possible choices both for the state and the age function are the following:
Power-law Exponential
State function g(w) (w/) , , 0 g(w) [exp(w/)1] , , , 0
Age function (t) (t / a)b , a,b 0 (t)a [exp(t/b)1] , a,b , ab0
Both these functional forms can reduce (or can tend) to a linear function. In particular:
the power-law function is linear if 1, the exponential function g(w) ( /)[exp(w/ ) 1] is linear for | | ,
thus allowing the TG model to reduce to a pure age-dependent (when g(w) is linear), or a pure state- dependent (when (t) is linear), or finally to a stationary process (both and (t) linear).
36 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
If g(w) (w/) , the mean degradation level and the variance of a new unit are in closed form, regardless of the functional form of (t):
[(t)1/ ] E{W (t)|0,0} (42) [(t)]
2 [(t)1/ ]2 V{W (t)|0,0} [(t)2/ ] . (43) [(t)] [(t)]
37 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Mean degradation Variance Variance-to-mean ratio
The behaviour of the mean, the variance, and the variance-to-mean ratio of the TG process with power-law (t) and g(w) functions, for selected b and values: b = 3, = 0.7, b = 3, = 3, b = 0.7, = 3, and b = 4, = 2. Large flexibility of the TG process in describing mean and variance behaviours.
38 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Numerical application
Let us consider the wear data of the 8 cylinder liners of the Diesel engine equipping a cargo ship of the Grimaldi Lines.
3.5
3.0
] 2.5
m
m
[
l 2.0
e
v
e
l
r 1.5
a
e Liner Liner
W 1.0 Liner Liner Liner Liner 0.5 Liner Liner
0.0 0 5 10 15 20 25 30 35 40 Operating time [103 hours]
39 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
3.5 0.20
] Gamma process 0.18 Gamma process
m 3.0
] 2
m
[ 0.16 m
r m
2.5 [
a 0.14
e
e c
w
n 0.12
n 2.0 a
TG process i
a r
e
a 0.10 v
m TG process
1.5 d
d 0.08 e
e t
t a a 0.06
1.0 m i
m t
i
t s
s 0.04 E
E 0.5 0.02 0.0 0.00 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Operating time [103 hours] Operating time [103 hours]
TG process with power-law functions (t)(t /a)b and g(w)(w/ ) (aˆ 5107 hours, bˆ1.701, ˆ 0.750 mm, ˆ 2.313, ˆ 0.59). Gamma process with power-law functions (t)(t/a)b (aˆ 562 hours, bˆ0.844, ˆ 0.0983 mm, ˆ 2.44). The likelihood ratio test suggests to reject the null hypothesis that the process is Gamma: 6.06 and pvalue0.014.
40 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The threshold limit is assumed to be wmax = 4 mm (the warranty limit).
The exceeding of the threshold limit does not generally produce a sudden break down of the engine, because it continues to operate even if the threshold limit is exceeded, but the “failure” occurrence reduces the performance of the engine and increases the consumption of fuel and motor oil (resulting in increased toxic emissions).
The unit is inspected at time t (age/operating time of the liner).
The liner maintenance is optimized based on the planned inspections. In particular, at each inspection:
if the current wear of the liner exceeds the threshold limit wmax = 4 mm, the liner is immediately replaced with a new one, otherwise, we can:
a) defer the substitution of the liner to the time t + of the next inspection, or b) replace now the used liner with a new one.
41 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The utility of a used liner at the future time t of the next inspection depends on the unknown
degradation level wt and is:
Total cost of new liner Conditional mean remaining life (including the cost of mounting) at the time t Mean life of a new liner
Positive utility (liner not failed) cL E{X |t ,wt }/E{T} wt wmax U (wt ; ) 0 wt wmax Null utility (threshold value reached) [(w w )/c]d w w t max t max Negative utility (liner failed)
Parameters of cost due to failure
Likewise, the utility of a new liner at the time of the next inspection is:
cL E{X | ,w }/E{T} w wmax U (w ; ) 0 w wmax d [(w wmax )/c] w wmax
42 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
The expected utility loss over the interval (t, t ) up to the next inspection associated to the action a): deferring the substitution of the liner to the time of the next inspection, is
wmax E{UL ( )} U (w ) U (w ; ) f (w|t,w ) dw U (w ; ) f (w|t,w ) dw a t W (t ) t W (t ) t wt wmax
Expected reduction of the liner value Expected cost due to failure of used liner (U (wt )cL E{X|t,wt }/E{T} is the current utility of the liner) (including the minus sign)
The expected utility loss over the interval up to the next inspection associated to the
action b): replacing now the used liner with a new one, is
wmax E{UL ( )} U (0) U (w ; ) f (w) dw U (w ; ) f (w) dw U (w ) b W ( ) W ( ) t 0 wmax
Expected reduction of the value of the new liner Residual value of the (U (0)cL is the utility of the new liner) used (substituted) liner Expected cost due to failure of new liner (including the minus sign)
43 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
-5 By assuming: cL = 10,000 €, c = 6.07·10 mm & d = 1.2, the differences i between the expected utility losses associated to actions a) deferring the substitution of the liner to a later time, and b) replacing now the liner with a new one, when the next inspection is planned after = 10,000 hours, under the TG process and the Gamma process are:
i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 i = 8 Age t [h] 31,270 21,970 16,300 28,000 37,310 24,710 16,620 16,300 i Wear wi [mm] 2.85 2.00 1.35 2.75 3.05 2.15 2.35 2.10 i under TG process [€] 3726 5883 7136 4144 2819 5441 5850 6292 i under Gamma process [€] 1933 5713 7447 2739 -896 5329 4428 5233
A positive difference i implies that the substitution of the liner i should be deferred
The proposed maintenance strategy leads to defer the replacement of all the liners under the TG model. On the contrary, under the (inadequate) Gamma process, the liner 5 should be substituted now, thus producing a unnecessary cost of 2819 €.
For = 15,000 hours, only liner #5 has to be substituted under the TG process (5 = 2951€), whereas under the Gamma process we should replace liners #1, 4 & 5.
44 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
References 1. M. Guida, F. Postiglione, G. Pulcini. A time-discrete extended Gamma process for deterioration processes. In Reliability, Risk and Safety (B. Ale, I. Papazoglou & E. Zio, Eds), Taylor & Francis (London) 2010, pp. 746–753. 2. M. Guida, F. Postiglione, G. Pulcini. A time-discrete extended gamma process for time-dependent degradation phenomena. Reliability Engineering and System Safety, 105 (9), 2012, pp. 73–79. 3. M. Guida, G. Pulcini. The Inverse Gamma process for modeling state-dependent deterioration processes. In Advances in Safety, Reliability and Risk Management (C. Bérenguer, A. Grall & C. Guedes Soares, Eds), Taylor & Francis (London) 2012, pp. 1128–1135. 4. M. Guida, G. Pulcini. The inverse Gamma process: a family of continuous stochastic models for describing state- dependent deterioration phenomena. Reliability Engineering and System Safety, 120, 2013, pp. 72-79. 5. M. Giorgio, M. Guida, G. Pulcini. A new class of Markovian processes for deteriorating units with state dependent increments and covariates. IEEE Transactions on Reliability, 64 (2), 2015, pp. 562-578. 6. M. Giorgio, M. Guida, G. Pulcini. A condition‐based maintenance policy for deteriorating units. An application to the cylinder liners of marine engine. Applied Stochastic Models in Business and Industry, 31 (3), 2015, pp. 339-348.
45 The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena 4-th AMMSI Project Workshop - Grenoble 2016
Thank you for your attention !
Merci beaucoup pour votre attention !
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